Hypocoercivity methods go back to Villani ([1]}), were further developed by Dolbeault, Mouhot and Schmeiser ([2]}) as well as Grothaus and Stilgenbauer ([3]}), and generalized by Grothaus and Wang ([4]}) to the case with weak Poincaré inequalities. Since then, the latter have been successfully used to derive concrete convergence rates
for solutions to degenerate Fokker-Planck partial differential equations, see for example [5]}, [6]}, [7]}, where last mentioned reference allows for velocity-dependent second-order coefficients. In order to obtain the rate of convergence to their stationary solution and to their equilibrium measure for solutions to degenerate stochastic differential equations, respectively, it is assumed that a Poincaré inequality exists for the measure induced by the potential giving the force. The same is necessary for the complementary result [8]}, which uses a Lyapunov function approach instead. For further complementary results using the Lyapunov function approach see [9]}, [10]} and references therein. In contrast, the earlier mentioned framework by Grothaus and Wang is based on weak Poincaré inequalities introduced by Wang and Röckner in [11]}, which exist under very weak conditions.
This allows for sub-exponential convergence rate estimates, but the applications worked out there still require constant second-order coefficients. We generalize their result to coefficients depending on the second component, while also slightly relaxing the assumptions in the constant case.
